   Chapter 8.3, Problem 6ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In each of 3-6, the relation R is an equivalence relation on A. As in Example 8.3.5, first find the specified equaivalence classes. Then state the number of distinct equivalenvce classes for R and list them. A = { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5 } R is defined on A as follows: For   all   x , y ∈ A ,         x R y     ⇔ 3 | ( x − y ) Equivalence classes:, , , 

To determine

To find the distinct equivalence classes of R.

Explanation

Given information:

The relation R is an equivalence relation on the set A. A={4,3,2,1,0,1,2,3,4,5}. R is defined on A as follows:                 For all x,yA, x R y  3|(xy).

Calculation:

A={4,3,2,1,0,1,2,3,4,5}R={(x,y)A×A|3( xy)}

Let us first group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group).

3|(xy) is true if and only if xy is a multiple of 3 if and only if x and y differ by 0,3,6 or 9 (as the elements are between -4 and 5, including).

Thus (xy)R when x and y differ by 0, 3, 6 or 9. First group: (4,4),(4,1),(4,2),(4,5)                   (1,4),(1,1),(1,2),(1,5)                   (2,4),(2,1),(2,2),(2,5)                 &#

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